Axisymmetric finite anti-plane shear of compressible nonlinearly elastic circular tubes

Polignone, Debra A.; Horgan, Cornelius O.

doi:10.1090/qam/1162279

Authors: Debra A. Polignone and Cornelius O. Horgan
Journal: Quart. Appl. Math. 50 (1992), 323-341
MSC: Primary 73G05; Secondary 73C50
DOI: https://doi.org/10.1090/qam/1162279
MathSciNet review: MR1162279
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Abstract: The axial shear problem for a hollow circular cylinder, composed of homogeneous isotropic compressible nonlinearly elastic material, is described. The inner surface of the tube is bonded to a rigid cylinder while the outer surface is subjected to a uniformly distributed axial shear traction and the radial traction is zero. For an arbitrary compressible material, the cylinder will undergo both a radial and axial deformation. These axisymmetric fields are governed by a coupled pair of nonlinear ordinary differential equations, one of which is second-order and the other first-order. The class of materials for which axisymmetric anti-plane shear (i.e., a deformation with zero radial displacement) is possible is described. The corresponding axial displacement and stresses are determined explicitly. Specific material models are used to illustrate the results.

J. E. Adkins, Some generalizations of the shear problem for isotropic incompressible materials, Proc. Cambridge Philos. Soc. 50 (1954), 334–345. MR 60991
A. E. Green and J. E. Adkins, Large elastic deformations, Clarendon Press, Oxford, 1970. Second edition, revised by A. E. Green. MR 0269158
A. E. Green and W. Zerna, Theoretical elasticity, 2nd ed., Clarendon Press, Oxford, 1968. MR 0245245
James K. Knowles, The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids, Internat. J. Fracture 13 (1977), no. 5, 611–639 (English, with French summary). MR 462075, DOI https://doi.org/10.1007/BF00017296
R. Abeyaratne and C. O. Horgan, Bounds on stress concentration factors in finite anti-plane shear, J. Elasticity 13 (1983), no. 1, 49–61. MR 708842, DOI https://doi.org/10.1007/BF00041313

A. H. Jafari, C. O. Horgan, and R. Abeyaratne, Finite anti-plane shear of an infinite slab with a traction-free elliptical cavity: bounds on the stress concentration factor, Internat. J. Non-Linear Mech. 19, 431–443 (1984)

C. O. Horgan and S. A. Silling, Stress concentration factors in finite antiplane shear: numerical calculations and analytical estimates, J. Elasticity 18 (1987), no. 1, 83–91. MR 899418, DOI https://doi.org/10.1007/BF00155438
James K. Knowles, On finite anti-plane shear for incompressible elastic materials, J. Austral. Math. Soc. Ser. B 19 (1975/76), no. 4, 400–415. MR 475116, DOI https://doi.org/10.1017/S0334270000001272
James K. Knowles, On finite anti-plane shear for incompressible elastic materials, J. Austral. Math. Soc. Ser. B 19 (1975/76), no. 4, 400–415. MR 475116, DOI https://doi.org/10.1017/S0334270000001272

R. W. Ogden, Non-linear Elastic Deformations, Ellis Horwood, Chichester, 1984

A. Mioduchowski and J. B. Haddow, Finite telescopic shear of a compressible hyperelastic tube, Internat. J. Non-Linear Mech. 9, 209–220 (1974)

M. G. Faulkner, Compressibility effects for nearly incompressible elastic solids, Appl. Sci. Res. 25, 328–336 (1972)

V. K. Agarwal, On finite anti-plane shear for compressible elastic circular tube, J. Elasticity 9, 311–319 (1979)

Debra A. Polignone and Cornelius O. Horgan, Pure torsion of compressible non-linearly elastic circular cylinders, Quart. Appl. Math. 49 (1991), no. 3, 591–607. MR 1121689, DOI https://doi.org/10.1090/qam/1121689

A. Ertepinar and G. Erarslanoglu, Finite anti-plane shear of compressible hyperelastic tubes, Internat. J. Engrg. Sci. 28, 399–406 (1990)

C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, Band III/3, Springer-Verlag, Berlin, 1965, pp. 1–602. MR 0193816
Fritz John, Plane elastic waves of finite amplitude. Hadamard materials and harmonic materials, Comm. Pure Appl. Math. 19 (1966), 309–341. MR 201113, DOI https://doi.org/10.1002/cpa.3160190306

P. J. Blatz, Application of finite elastic theory in predicting the performance of solid propeliane rocket motors, Calif. Inst. of Tech., GALCIT SM60-25, 1960

P. J. Blatz, Application of large deformation theory to the thermo-mechanical behavior of rubberlike polymers—porous, unfilled, and filled, Rheology, Theory and Applications, vol. 5 (F. R. Eirich, ed.), Academic Press, New York, 1969, pp. 1–55

R. W. Ogden, Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids, Proc. Roy. Soc. London Ser. A 328, 567–583 (1972)

J. B. Haddow and R. W. Ogden, Compression of bonded elastic bodies, J. Mech. Phys. Solids 36 (1988), no. 5, 551–579. MR 964175, DOI https://doi.org/10.1016/0022-5096%2888%2990010-5

R. M. Christensen, A two material constant, nonlinear elastic stress constitutive equation including the effect of compressibility, Mech. of Materials 7, 155–162 (1988)

P. J. Blatz and W. L. Ko, Application of finite elasticity to the deformation of rubbery materials, Transactions of the Society of Rheology 6, 223–251 (1962)

M. Levinson and I. W. Burgess, A comparison of some simple constitutive relations for slightly compressible rubber-like materials, Internat. J. Mech. Sci. 13, 563–572 (1971)

Y. C. Fung, Biomechanics. Mechanical properties of living tissues, Springer-Verlag, Berlin, 1981

M. F. Beatty, Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues— with examples, Appl. Mech. Reviews 40, 1699–1734 (1987)

P. F. Chu, Strain energy function for biological tissues, J. Biomechanics 3, 547–550 (1970)

Qing Jiang and James K. Knowles, A class of compressible elastic materials capable of sustaining finite anti-plane shear, J. Elasticity 25 (1991), no. 3, 193–201. MR 1115552, DOI https://doi.org/10.1007/BF00040926